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Elsevier Editorial System(tm) for International Journal of Multiphase Flow Manuscript Draft Manuscript Number: Title: Microstructural effects on the permeability of periodic fibrous porous media Article Type: Original Research Paper Keywords: Permeability; Fibrous porous media; FEM; Drag relations; Carman-Kozeny equation; Incompressible fluids. Corresponding Author: Mr. Kazem Yazdchi, Corresponding Author's Institution: University of Twente First Author: Kazem Yazdchi Order of Authors: Kazem Yazdchi; S. Srivastava , PhD; S. Luding, PhD Abstract: An analytical-numerical approach is presented for computing the macroscopic permeability of fibrous porous media taking into account their micro-structure. A finite element (FE) based model for viscous, incompressible flow through a regular array of cylinders/fibers is employed for predicting the permeability associated with this type of media. High resolution data, obtained from our simulations, are utilized for validating the commonly used semi-analytical models of drag relations from which the permeability is often derived. The effect of porosity, i.e., volume fraction, on the macroscopic permeability is studied. Also micro-structure parameters like particle shape, orientation and unit cell staggered angle are varied. The results are compared with the Carman-Kozeny (CK) equation and the Kozeny factor (often assumed to be constant) dependence on the micro-structural parameters is reported and used as an attempt to predict a closed form relation for the permeability in a variety of structures, shapes and wide range of porosities.
Professor A. Prosperetti Editor-in-Chief International Journal of Multiphase Flow Department of Mechanical Engineering, The Johns Hopkins University, 119 Latrobe Hall, Baltimore, MD 21218, USA
19th February 2011
Dear professor A. Prosperetti,
Re: Microstructural effects on the permeability of periodic fibrous porous media K. Yazdchi, S. Srivastava and S. Luding Multi Scale Mechanics (MSM), Faculty of Engineering Technology, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands I am submitting the above titled manuscript for review and possible publication in the International Journal of Multiphase Flow journal. Prediction of the permeability of fibrous porous media is a longstanding but still challenging engineering problem, with many practical applications. Here, we consider periodic arrays of parallel cylinders perpendicular to the flow direction and studied with a finite element (FE) based model. We believe that permeability/drag data obtained in this study can be used as a design guide in composites manufacturing and utilized for validation of advanced, more coarse-grained models for particle-fluid interactions.
This manuscript is submitted for publication in International Journal of Multiphase Flow. It has not been previously published, is not currently under review for publication in any other journal and will not be submitted elsewhere during peer review in this journal. I shall be looking forward to hearing from you. Yours sincerely, K. Yazdchi
Cover Letter
1
Microstructural effects on the permeability of periodic fibrous porous media
K. Yazdchi, S. Srivastava and S. Luding
Multi Scale Mechanics (MSM), Faculty of Engineering Technology,
University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands
Abstract
An analytical-numerical approach is presented for computing the macroscopic
permeability of fibrous porous media taking into account their micro-structure. A finite
element (FE) based model for viscous, incompressible flow through a regular array of
cylinders/fibers is employed for predicting the permeability associated with this type of
media. High resolution data, obtained from our simulations, are utilized for validating the
commonly used semi-analytical models of drag relations from which the permeability is
often derived. The effect of porosity, i.e., volume fraction, on the macroscopic
permeability is studied. Also micro-structure parameters like particle shape, orientation
and unit cell staggered angle are varied. The results are compared with the Carman-
Kozeny (CK) equation and the Kozeny factor (often assumed to be constant) dependence
on the micro-structural parameters is reported and used as an attempt to predict a closed
form relation for the permeability in a variety of structures, shapes and wide range of
porosities.
Keywords: Permeability; Fibrous porous media; FEM; Drag relations; Carman-Kozeny
equation; Incompressible fluids.
*ManuscriptClick here to view linked References
2
1. Introduction
The problem of creeping flow between solid bodies arranged in a regular array is
fundamental in the prediction of seepage through porous media and has many
applications, including: composite materials [10, 19], rheology [21, 20], geophysics [3],
polymer flow through rocks [27], statistical physics [14, 7], colloid science [26], soil
science [23, 8] and biotechnology [33]. A compelling motivation for such studies
concerns the understanding, and eventually the prediction, of single and multiphase
transport properties of the pore structure.
Usually, when treating the medium as a continuum, satisfactory predictions can be
obtained by Darcy's law, which lumps all complex interactions between the fluid and
fibers/particles into K, the permeability (tensor). Accurate permeability data, therefore, is
a critical requirement for macroscopic simulations based on Darcy’s law – to be
successfully used for design and optimization of industrial processes.
The Ergun equation is a semi-empirical drag relation from which the permeability of
porous media can be deduced. It is obtained by the direct superposition of two asymptotic
solutions, one for very low Reynolds number, the Carman-Kozeny (CK) equation [4],
and the other for very high Reynolds numbers, the Forchheimer correction [4]. However,
these approximations do not take into account the micro-structural effects, namely the
shapes and orientations of the particles, such that not only local field properties but also
some global properties (such as anisotropy) cannot be addressed.
In this respect, two distinct approaches seem to have emerged. The first approach is based
on lubrication theory and considers the pores of a porous medium as a bunch of capillary
tubes which are tortuous or interconnected in a network [4]. Even though this model has
been used successfully for isotropic porous media, it does not work well for either axial
or transverse permeability of aligned fibrous media [5].
The second approach (cell method) considers the solid matrix as a cluster of immobile
solid obstacles, which collectively contribute Stokes resistance to the flow. For a review
of these theories see [10, 4]. When the solids are dilute, i.e., at high porosities, basically
the particles do not feel each other, so that cell approach is appropriate. Bruschke and
Advani [5] used lubrication theory in the high fiber volume fraction range but adopted an
3
analytical cell model for lower fiber volume fractions. A closed form solution, over the
full fiber volume fraction range, is obtained by matching both solutions asymptotically.
Prediction of the permeability of fibrous media dates back to experimental work of
Sullivan [28] and theoretical works of Kuwabara [17], Hasimoto [13], and Happel [12].
The parallel flow solutions are idealized solutions for the flow through cigarette filters,
plant stems and around pipes in heat exchange tanks. The transverse solutions are
applicable to transverse fibrous filters used for cleaning liquids and gases and regulating
their flow. Both types of solutions can also be applicable to the settling of suspensions of
long thin particles. A comprehensive review of experimental works of permeability
calculation of these systems is available in Jackson et al. [15] and Astrom et al. [2].
Later, Sangani and Acrivos [25], performed analytical and numerical studies of the
viscous permeability of square and staggered arrays of cylinders. Their analytical models
were accurate in the limits of low and high porosity. For high densities they obtained the
lubrication type approximations for narrow gaps. Drummond and Tahir [9] modeled the
flow around a fiber using a unit cell approach (by assuming that all fibers in a fibrous
medium experience the same flow field) and obtained equations that are applicable at
lower volume fractions. Gebart [11] presented an expression for the longitudinal flow,
valid at high volume fractions, that has the same form as the well-known KC equation.
For transverse flow, he also used the lubrication approximation, assuming that the narrow
gaps between adjacent cylinders dominate the flow resistance. Using the Eigen-function
expansions and point match methods, Wang [32] studied the creeping flow past a
hexagonal array of parallel cylinders.
Our literature survey indicates that relatively little attention has been paid to the
macroscopic permeability of ordered periodic fibrous materials. More importantly, the
majority of the existing correlations for permeability are based on curve-fitting of
experimental or numerical data and most of the analytical models found in the literature
are not general and fail to predict permeability over the wide range of porosity, since they
contain some serious assumptions that limit their range of applicability.
In this study, periodic arrays of parallel cylinders (with circular, ellipse and square cross-
section) perpendicular to the flow direction are considered and studied with a FE based
model in section 2. The effects of shape and orientation as well as porosity and structure
4
on the macroscopic permeability of the porous media are discussed in detail. In order to
relate our results to available work, the data are compared with previous theoretical and
numerical data for square and hexagonal packing configurations and a closed form
relation is proposed in section 3 in the attempt to combine our various simulations. The
paper is concluded in section 4 with a summary and outlook for future work.
2. Results from FE simulations
This section is dedicated to the FE based model simulations and the results on
permeability as function of porosity, structure, shape and anisotropy.
2.0. Introduction and Terminology
The superficial velocity, U, within the porous media in the unit cell is defined as
uudvV
U
fV
1
, (1)
where u, u , V, Vf and are the local microscopic velocity of the fluid, corresponding
averaged velocity, total volume, volume of the fluid and porosity, respectively. For the
case where the fluid velocity is sufficiently small (creeping flow), the well-known
Darcy’s law relates the superficial fluid velocity U through the pores with the pressure
gradient, p , measured across the system length, L, so that
pK
U
, (2)
where μ and K are the viscosity of the fluid and the permeability of the sample,
respectively. At low Reynolds numbers, which are relevant for most of the composite
manufacturing methods, the permeability depends only on the geometry of the pore
structure.
Recently, models based on Lagrangian tracking of particles combined with computational
fluid dynamics for the continuous phase, i.e., discrete particle methods (DPM), have
become state-of-the-art for simulating gas-solid flows, especially in fluidization processes
5
[16]. In this method, two-way coupling is achieved via the momentum sink/source term,
Sp which models the fluid-particle drag force
p pS u v , (3)
where the interphase momentum-transfer coefficient, , describes the drag of the
gas/fluid phase acting on the particles and vp is the velocity of particles. (Additional
effects like the added mass contributions are disregarded here for the sake of simplicity.)
In steady state, without acceleration, wall friction, or body forces like gravity, the fluid
momentum balance reduces to
0pp u v . (4)
By comparing Eqs. (2) and (4), using the definition of Eq. (1), and assuming immobile
particles, i.e., 0pv , the relation between and permeability K is
K
2 . (5)
Accurate permeability data, therefore, is a critical requirement in simulations based on
DPM to be successfully used in the design and optimization of industrial processes.
In the following, results on the permeability of two-dimensional (2D) regular periodic
arrays of cylinders with different cross section are obtained by incorporating detailed FE
simulations. This is part of a multiscale modeling approach and will be very useful to
generate closure or coupling models required in more coarse-grained, large-scale models.
2.1. Mathematical formulation and boundary conditions
Both hexagonal and square arrays of parallel cylinders perpendicular to the flow direction
are considered, as shown in Fig. 1. The basis of such model systems lies on the
assumption that the porous media can be divided into representative volume elements
(RVE) or unit cells. The permeability is then determined by modeling the flow through
one of these, more or less, idealized cells. At the left and right pressure- and at the top
and bottom periodic-boundary conditions are applied. The no-slip boundary condition is
applied on the surface of the particles/fibers. A typical unstructured, fine and triangular
FE mesh is also shown in Fig. 1(c). The mesh size effect is examined by comparing the
simulation results for different resolutions (data are not shown here). The range of
6
number of elements is varying from 10000 to 100000 depending on the porosity regime.
It should be noted that in Darcy’s linear regime (creeping flow) – although we have
applied pressure boundary conditions at left and right – identical velocity profile at inlet
and outlet are observed, due to the symmetry of this geometry and linearity. However, by
increasing the pressure gradient (data not shown), the flow regime changes to non-linear
and becomes non-symmetric. Furthermore, because of the symmetry in the geometry and
boundary conditions, the periodic boundary condition and symmetry boundary condition,
i.e., zero velocity in vertical direction at top and bottom of the unit cells, will lead to
identical results (as confirmed by simulations – data not shown).
(a) (b)
(c)
Figure 1: The geometry of the unit cells used for (a) square and (b) hexagonal
configurations, with angles 450 and 60
0 between the diagonal of the unit-cell and the
horizontal flow direction (red arrow), respectively. (c) shows a typical quarter of an
unstructured, fine and triangular FE mesh.
Flow direction
7
2.2. Permeability of the square and hexagonal arrays
Under laminar, steady state condition, with given viscosity, the flow through porous
media is approximated by Darcy’s law. By calculating the superficial velocity, U, from
our FE simulations and knowing the pressure gradient, p , over the length of the unit
cell, L, we can calculate the dimensionless permeability (normalized by the cylinder
diameter, d), K/d2. In Table 1, various correlations from the literature are listed. The first
relation by Gebart [11] has a peculiar analytical form and is valid in the limit of high
density, i.e., low porosity – close to the close packing limit c (the same as Bruschke [5]
in the low porosity limit, with maximum discrepancy less than 1%). Note that the
relations by Happel [12], Drummond et al. [9], Kuwabara [17], Hasimoto [13], and
Sangani et al. [25] are all identical in their first term – that is not dependent on the
structure – in the limit of small solid volume fraction , i.e., large porosity. In contrast,
their second term is weakly dependent on the structure (square or hexagonal). Bruschke
et al. [5] proposed relations that are already different in their first term. The last two
relations in the table are only valid in intermediate porosity regimes and do not agree
with any of the above relations in either of the limit cases.
In Fig. 2, the variation of the (normalized) permeability, K/d2, with porosity, for square
and hexagonal packings is shown. The lubrication theory presented by Gebart [11] agrees
well with our numerical results at low porosities ( 0.6 ), whereas, at high porosities
( 0.6 ), the prediction by Drummond [9] better fits our data. Drummond et al. [9] have
found the solution for the Stokes equations of motion for a viscous fluid flowing in
parallel or perpendicular to the array of cylinders by matching a solution outside one
cylinder to a sum of solutions with equal singularities inside every cylinder of an infinite
array. This was in good agreement with other available approximate solutions, like the
results of Kuwabara [17] and Sangani et al. [25] at high porosities, as also confirmed by
our numerical results (data not shown). Note that our proposed merging function in
section 3.4, fits to our FE results within 2% error for the whole range of porosity.
8
0.2 0.4 0.6 0.8 110
-6
10-4
10-2
100
102
K/ d
2
Gebart [11]
Drummond et. al. [9]
FE simulations
0.2 0.4 0.6 0.8 110
-6
10-4
10-2
100
102
K/d
2
Gebart [11]
Drummond et. al. [9]
FE simulations
Figure 2: Normalized permeability plotted against porosity for (a) square and (b)
hexagonal packing for circular shaped particles/cylinders with diameter d, for
perpendicular flow. The lines give the theoretical predictions, see inset. For high
porosities, the difference between Gebart and Drummond, in the hexagonal
configuration, is less than 5%, while for the square configuration it is less than 30%.
(a) Square configuration
(b) Hexagonal configuration
9
Table 1: Summary of correlations between normalized permeability, K/d2 and porosity,
with 1 , the solid volume fraction.
Author K/d2
Comments
Gebart [11]
5/ 2
11
1
cC
4, 1 / 4
9 2
4, 1 / 2 3
9 6
c
c
C
C
Square configuration: 2s
GK d
Hexagonal config.: 2h
GK d
Bruschke
et al. [5]
1
12
2 2
3 2
1tan
1 13 1
12 21
l
l l ll
l l
Lubrication theory, square
config.:
142l
Drummond et
al. [9]
2
2
1 1 2 0.796ln 1.476
32 1 0.489 1.605
2 541 1 2.534
ln 1.497 2 0.73932 2 1 1.2758
Square configuration: 2s
DK d
Hexagonal config.: 2h
DK d
Bruschke
et al. [5]
2
2
1 1 3ln 2
32 2 2
Cell method, square config.
Kuwabara
[17]
21 1ln 1.5 2
32 2
Based on Stokes approximation
Hasimoto
[13]
1 1ln 1.476
32
Using elliptic functions:
21 1ln 1.476 2
32O
Sangani et al.
[25]
2 31 1ln 1.476 2 1.774 4.076
32
Happel [12]
2
2
1 1 1ln
32 1
Lee and Yang
[18]
3
1.3
0.2146
31 1
Valid for 0.435 0.937
Sahraoui et al.
[24]
5.10.0152
1
Valid for 8.04.0
10
2.3. Effect of shape on the permeability of regular arrays
In this subsection, we investigate the anisotropic behavior of permeability due to particle
shape in square configuration. Using elementary algebraic functions, Zhao et al. [34]
derived the analytical solutions for pore-fluid flow around an inhomogeneous elliptical
fault in an elliptical coordinate system. Obdam and Veling [22] employed the complex
variable function approach to derive the analytical solutions for the pore-fluid flow within
an elliptical inhomogeneity in a two-dimensional full plane. Zimmerman [36] extended
their solutions to a more complicated situation, where a randomly oriented distribution of
such inhomogeneous ellipses was taken into account. Wallstrom et al. [31] later applied
the two-dimensional potential solution from an electrostatic problem to solve a steady-
state pore-fluid flow problem around an inhomogeneous ellipse using a special elliptical
coordinate system. More recently, Zhao et al. [35] used inverse mapping to transform
those solutions into a conventional Cartesian coordinate system.
Here, in order to be able to compare different shapes and orientations, the permeability
will be normalized with respect to the obstacle length, Lp, which is defined as
Lp = 4 area / circumference
Lp = 2r = d (for circle), Lp = c (for square), Lp = 4πab / AL (for ellipse) (6)
where r, c, a and b are the radius of the circle, the side-length of the square, the major
(horizontal) and minor (vertical) length of the ellipse, respectively. AL is the
circumference of the ellipse.
By applying the same procedure as in the previous section, the normalized permeability
(with respect to obstacle length, Lp) is calculated for different shapes on a square
configuration.
In Fig. 3 the normalized permeability is shown as function of porosity for different
shapes. At high porosities the shape of particles does not affect much the normalized
permeability, but at low porosities the effect is more pronounced. We observe that circles
have the lowest and horizontal ellipses the highest normalized permeability.
11
0.2 0.4 0.6 0.8 110
-6
10-4
10-2
100
102
K/L
p 2
Circle
Square
Ellipse
Figure 3: Effect of shape on the normalized permeability from a square packing
configuration of circles, squares and ellipses (a/b=2, major axis in flow direction). The
lines are only connecting the data-points as a guide to the eye.
2.4. Effect of aspect ratio on the permeability of regular arrays of ellipses
In this subsection the effect of aspect ratio, a/b on the normalized permeability of square-
arrays of ellipses is investigated. In fact, the case of high aspect ratio at high porosity
represents the flow between parallel plates (slab flow). The relation between average
velocity, u , and pressure drop for slab flow is
2
12
sh pu
L
(7)
where hs is the distance between parallel plates (in our square configuration, in the limit
/ 1a b , one has hs=L=1). Note that, since there are no particles, 1 , the average and
superficial velocities are identical, i.e., u U . By comparing Eqs. (7) and (2) the
permeability, i.e., 2 /12sK h is obtained, which indeed shows the resistance due to no
slip boundaries at the walls. The variation of permeability for a wide range of aspect
ratios at different porosities is shown in Fig. 4. It is observed (especially at high
12
porosities) that by increasing the aspect ratio the permeability increases until it reaches
the limit case of slab flow for which the permeability is 2 /12 0.0833sK h [m
2]. The
aspect ratio b/a <1 means that the ellipse stands vertically and therefore the permeability
reduces drastically.
10-2
10-1
100
101
102
10-8
10-6
10-4
10-2
100
b/a
K [m
2]
= 0.6
= 0.7
= 0.8
= 0.9
= 0.95
Slab flow (hs
2 /12 [m
2])
Figure 4: Effect of aspect ratio on the permeability of square configurations of ellipses
with different porosities as given in the inset. The lines are only connecting the data-
points as a guide to the eye.
2.5. Effect of orientation on the permeability of regular arrays
By changing the orientation (θ), i.e., the angle between the major axis of the obstacle and
the horizontal axis, not only the values of the permeability tensor will change, but also its
anisotropy will show up (so that the pressure gradient and the flow velocity are not
parallel anymore). Therefore, the geometry of the pore structure has great effect on the
permeability in irregular fibrous media. This effect for squares and ellipses (a/b = 2) is
shown in Fig. 5.
13
For square shapes, at high porosities, the orientation does not much affect the
permeability, whereas at low porosities the permeability depends a lot on the orientation.
At θ = 450 we observe a drop in permeability, because we are close to the blocking
situation, i.e., zero permeability, at a critical porosity (at which the permeability drops to
zero) of 2
11 0.5
2sin 45c
.
For ellipses, at high porosity, the orientation does not affect the permeability, whereas at
low porosities the effect is strong. By increasing the orientation angle, i.e., by turning the
major axis from horizontal to vertical, the permeability is reduced. The critical porosity
εc≈0.6073 is purely determined by the major axis of the ellipse for θ = 900.
14
0.2 0.4 0.6 0.8 110
-5
10-4
10-3
10-2
10-1
100
101
K/L
p 2
= 0o
= 10o
= 20o
= 30o
= 45o
(a)
0.2 0.4 0.6 0.8 1
10-4
10-5
10-3
10-2
10-1
100
101
K/L
p 2 = 0o
= 30o
= 45o
= 60o
= 90o
(b)
Figure 5: Effect of orientation (θ) on the normalized permeability for (a) square and (b)
ellipse (a/b=2) in square packing configurations at different porosity. The lines are only
connecting the data-points as a guide to the eye.
15
The general form of Darcy’s law for anisotropic media in 2D in matrix form can be
written as
11 12
21 22
1x
y
pU K K x
pK KU
y
, (8)
where <Ux> and <Uy> are superficial velocities in x and y direction, respectively. Then
the permeability tensor for any value of can be calculated as
011 12
9021 22
0
0
TKK K
K R RKK K
(9)
where 0K and 90K are the principal values of permeability that are determined from the
values of 00 and 090 , respectively. In Eq. (9), RT is the transposed of the rotation
matrix R, defined as (counterclockwise rotation by )
cossin
sincosR (10)
Eq. (9) shows that for 00 90,0 , one has 012 K , which means that by applying a
pressure gradient in x direction, one gets a superficial velocity in y direction (i.e.
anisotropic behavior because of oriented shape). Our numerical results are in good
agreement with theoretical predictions (Eq. (9)) especially at high porosities (see the solid
lines in Fig. 6). We have more deviation at low porosities (maximum discrepancy ≈ 5%)
because of channel blockage and changes in flow behavior (the comparison is not shown
here).
In Fig. 6, the variation of normalized permeability is shown as a function of the
orientation angle . The normalized permeability is symmetric to 900 and decreases by
increasing the orientation angle from 00. The eigenvalues of the permeability tensor are
the extrema of the curves and the other data are well fitted by
2
0 90 0 902 cos 2 2pK L K K K K . By decreasing the aspect ratio a/b, we
approach the value for the circular (cylinder) obstacle, i.e., a/b =1. The normalized
permeability is symmetric to 450 for square shapes.
16
0 20 40 60 80 100 120 140 160 1800
0.05
0.1
0.15
0.2
0.25
0.3
0.35
[degree]
K/L
p2
Figure 6: Normalized permeability plotted against orientation angle for different shapes
at porosity 85.0 , for different obstacle shapes in square configurations. The dashed
line represents circles. The solid lines show the theoretical predictions according to Eq.
(9), where the eigenvalues are taken from the 00, 90
0 and 0
0, 45
0 degrees for ellipses and
squares simulations, respectively.
2.6. Effects of staggered cell angle
In this subsection, the effect of another micro-structural parameter, the staggered cell
angle, , on the normalized permeability for circles (Lp=d) is discussed. The staggered
angle is defined between the diagonal of the unit-cell and flow-direction (horizontal), see
Fig. 7. In addition to the special cases o45 and o60 , i.e., square and hexagonal
packings, respectively, several other angles are studied. The contour of the horizontal
velocity field component, for different , at constant porosity 0.7 , is shown in Fig.
7. By changing , the flow path and also the channel length will change. At 070 and
higher, the flow mainly follows a straight line, indicated by arrows in Fig. 7(a), with large
superficial velocity and consequently large values of permeability. However, by
decreasing down to 350, the flow pattern completely changes and the superficial
17
velocity reduces, which should lead to lower and lower permeability. In brief, with
increasing angle, both the superficial velocity and the permeability increase, with a
plateau around o45 .
(a) (b)
(b) (d)
Figure 7: Horizontal velocity field components for (a) 060 , (b)
050 , (c)
040 and (d) 035 at fixed porosity ε = 0.7. The arrows indicate the main flow
channel in (a) and (b). The staggered angle is defined between the diagonal of the unit-
0 0.03 0.06 0.09 0.12 0.15 0.18 0.21 0.24 [m/s]
18
cell and flow-direction (horizontal). The minimal angle 1 0
min
2 1tan 10.81
is realized when the vertical opening is closed, while the maximal angle
1 0
max tan 79.182 1
corresponds to the closed horizontal pore.
In Fig. 8 the normalized permeability is shown as function of the staggered angle, , at
different porosities. As it is seen the arrangement of particles relative to the flow
direction is important in determining the permeability. By increasing , the normalized
permeability increases (the vertical distance between particles increases and therefore the
resistance to the flow decreases) until it reaches a local maximum at 035 –
consistently for different porosities. At larger angles, it slightly decreases and attains a
local minimum at 055 , beyond which it increases rapidly again. This behavior can be
explained by the variation of the area-fraction distribution with on the planes
perpendicular to the flow direction, as discussed in [1].
The normalized permeability as a function of can be expressed as a cubic polynomial
3 20 0 0
2 0 0 0
45 45 45A B C D
45 45 45
K
d
(11)
where A, B, C and D are constants, listed in Table 2 for different porosities.
19
20 30 40 50 60 700
0.02
0.04
0.06
0.08
0.1
0.12
0.14
[degree]
K/d
2
Figure 8: The variation of normalized permeability versus staggered unit cell angle at
different porosities, as given in the inset. The solid lines show the fit (Eq. (11))
Table 2: Fitted parameters for the permeability-staggered angle relation
8.0 7.0 6.0
A 0.38401 0.18813 0.09913
B 0.02682 0.01738 0.01171
C -0.03693 -0.01914 -0.01027
D 0.07698 0.02584 0.009319
The leading term with A is dominating, the term with B is a rather small correction, the
term with C sets the (negative) slope in the center, and the term with D determines the
offset. All fit-parameters depend on porosity and it should be noted that the range of
available angles is limited and also depends on porosity. Additional scaling- and fit-
attempts (data not shown) did not lead to much better results, thus we only present this
empirical fit here.
The decreasing region, i.e., 0 035 55 corresponds to the case in which the flow goes
in a preferred channel orthogonal/perpendicular to the line (diagonal) connecting two
particles, see Figs. 7(b) and (c). While in cases with larger , the flow goes at straight
20
lines/channels, see Fig. 7(a), the configuration for smaller is dominated by the narrow
vertical opening between two obstacles. In essence, in the plateau region, the
permeability is not much affected by the staggered angle . This observation might be
useful during design and manufacturing of fibrous composites.
In summary, the results of this section show that the macroscopic permeability not only
depends on the porosity but also on the microstructure, namely shape, aspect ratio and
orientation of particles.
3. Theoretical prediction of the permeability for all porosities
In this section, based on the observations in the previous section, and using the velocity
profiles from FE unit cell simulations, a generalized form of the Carman Kozeny (CK)
equation for the permeability of fibrous porous media (2D regular arrays of particles) is
proposed.
3.1 From special cases to a more general CK equation
The earliest and most widely applied approach in the porous media literature, for
predicting the permeability, involves capillary models [6] such as the one that leads to the
CK equation. The approach to obtain this equation is based on Poiseuille flow through
pipes. Assuming pipe flow through a cylindrical channel of diameter hp, the average
velocity through the channel is
2
32
p
p
h pu
L
, (12)
and for slab flow through an infinite channel of width hs
2
12
ss
h pu
L
, (13)
given the pressure drop p per length L, and a fluid with viscosity .
Defining the hydraulic diameter
volume available for flow4
total wetted surfacehD , (14)
21
allows to unify the relations above, by combining Eq. (14) with either Eq. (12), with
h pD h , or Eq. (13), with 2h sD h , and with Darcy’s law, Eq. (2), so that the
permeability is described by the CK relation [6]
2
h
CK
u DK
p
(15)
where CK =32 (or 48) is the dimensionless Kozeny factor, characteristic of the pipe (or
slab) pore structure. When one has obstacles like fibers (or particles) instead of straight
pores, the hydraulic diameter can be re-written as
4 4 particle surface 4
, with1 1 particle volume 1
vh v
v v
SV dD a
S a V d
, (16)
with the total volume of the unit cell, V, the total wetted surface, Sv, the specific surface
area, av, and the porosity, , for a fibrous medium of fiber diameter d. Note that the
hydraulic diameter, in this way, is expressed as a function of the measurable quantities
porosity and specific surface area. The above value of av is for circles (cylinders) – for
spheres one has av=6/d. In this formulation, we just consider the resistance due to
presence of particles (no slip boundaries at the surface of the particles) and neglect the
outer walls.
Inserting Eq. (16) into Eq. (15), yields the normalized permeability for fibers
3
22
1
1CK
K
d
, (17)
which depends non-linearly on the porosity and on a shape/structure factor, CK .
One of the main drawbacks of the CK equation is that the Kozeny factor CK is a-priori
unknown in realistic systems and has to be determined experimentally. An ample amount
of literature exists on the experimental and theoretical determination of the Kozeny
factor, but we are not aware of a theory that relates CK with the microstructure, i.e., the
porosity, the random configuration, tortuosity, and other microscopic quantities. An
overview of experimental and theoretical approaches can be found in Ref. [2], which
mainly deals with fibrous media, and Ref. [29], which is based on variational principles.
One of the most widely accepted approaches to generalize the CK relation was proposed
22
by Carman [6], who noticed that the streamlines in a porous medium are far from being
completely straight and parallel to each other. This effect can be described by a
dimensionless parameter, Le/L (tortuosity), with the length of the streamlines, Le, relative
to the length of the sample, L. Hence the Kozeny factor can be split into
2
eCK
L
L
(18)
where is the effect of particle shape, which can be seen as a fitting parameter. In fact,
the tortuosity and the shape factor reflect the effects of microstructure on the macroscopic
properties (like permeability) of the porous media.
In the original form of the CK equation for random 3D sphere structures, it is assumed
that the tortuosity is a constant for all ranges of porosities and is equal to 2 and the
fitting parameter, , then becomes 90 for the case of pipe flow and 60 for slab flow.
Knowing the values of the normalized permeability from our FE simulations, we can
compare the values of the Kozeny factor based on our numerical results and available
theoretical data and with the original CK =120 for slab flow, see Eqs. (17) and (18). The
comparison is shown in Fig. 9. At a certain range of porosities, 7.05.0 , the CK
factor CK is indeed not varying much. However, at higher or lower porosities, it
strongly depends on porosity and structure. At high and low porosities, our numerical
results are in good agreement with the predictions of Drummond [9] and Gebart [11],
respectively (see Table 1). These results indicate that the Carman-Kozeny factor, CK , is
indeed not constant and depends on the microstructure.
In the following subsections, we will study the dependency of CK on the micro-
structural parameters.
23
0.2 0.4 0.6 0.8 10
100
200
300
400
500
600
K
C
Original CK
FE simulation (square)
FE simulation (hexagonal)
Gebart [11] (hexagonal)
Gebart [11] (square)
Drummond [9] (hexagonal)
Drummond [9] (square)
Figure 9: Kozeny factor plotted as a function of porosity for different models (lines) and
data sets (symbols) as given in the inset.
3.2. Measurement of the tortuosity (Le/L)
As discussed before, the tortuosity is the average effective streamline length scaled by
system length, Le/L, and one possible key parameters in the Kozeny factor in the CK
equation [6]. From our numerical simulations, we extract the average length of several
streamlines (using 8 streamlines that divide the total mass in-flux into 9 zones, thus
avoiding the center and the edges). By taking the average length of these lines, the
tortuosity can be obtained, while by taking the standard deviation of the set of
streamlines, the homogeneity of the flow can be judged. The tortuosity is plotted in Fig.
10 as function of porosity for different shapes and orientations and as function of the
staggered angle for different porosities.
Unlike the traditional form of the CK equation, which assumes that / 2eL L (for
random 3D structures) is constant [6], our numerical results show that the tortuosity (i) is
smaller and (ii) depends on the porosity and the pore structure. In Fig.10(a), as intuition
suggests, the vertical and horizontal ellipses have the highest and lowest average
24
tortuosity, respectively. This goes ahead with very large and very small standard
deviation, i.e., the vertical ellipse configuration involves the widest spread of streamline
lengths. In the case of a circular shape obstacle, the (average) tortuosity is between the
horizontal and vertical ellipse cases and, for intermediate porosity, becomes almost
independent of porosity, with constant standard deviation that is wider than the average
tortuosity deviation from unity. The square shape obstacles are intermediate in tortuosity,
i.e., the 00-square (45
0-square) shapes take tortuosity values between the horizontal
(vertical) ellipse and the circular obstacles.
In Fig. 10(b), the tortuosity is plotted against the staggered unit cell angle . By
increasing the value of tortuosity increases until about 045 , where it reaches its
maximum. Note that the standard deviation remains small for all angles smaller than
045 . At higher angles, tortuosity decreases, while its standard deviation considerably
increases. At large values of , most of the fluid flow goes along a straight line,
however, near the boundary, we have a few longer streamlines that cause the large
standard deviation. At the limit case of max , when the upper particles touch, see Fig.
7, the tortuosity approaches unity (data not shown) and the flow goes mostly along a
straight channels.
3.3. Measurement of the shape/fitting factor ( )
Knowing the values of tortuosity, Le/L, and normalized permeability, K/d2, from our FE
simulations, we can obtain the values of for different shapes and orientations as
2
eCK
L
L
. In Fig. 11 the variation of as a function of porosity for different
shapes and orientations is shown. Unlike the traditional CK factor, the shape/fitting factor
is not only a function of porosity but also depends on the orientation of
particles/cylinders. This dependency is more pronounced at high (low) porosities and
close to the blocking conditions, i.e., ellipses with 900 and squares with 45
0.
25
0.4 0.5 0.6 0.7 0.8 0.9 10.98
1
1.02
1.04
1.06
1.08
1.1
1.12
1.14
1.16
Le/L
Ellipse-90o
Square-45o
Circle
Square-00
Ellipse-00
(a)
10 20 30 40 50 60 70 801
1.05
1.1
1.15
1.2
1.25
[degree]
Le/L
= 0.6
= 0.7
= 0.8
(b)
Figure 10: Tortuosity LLe / (a) plotted as a function of porosity for different obstacle
shapes and orientations on square configurations, o45 , (b) plotted as a function of the
staggered cell angle, , as given in the inset, for circles at different porosities on
hexagonal configurations. Error bars give the standard deviation of the 8 streamline
26
lengths, where bottom-values below unity indicate a highly non-symmetric distribution
around the average.
0.4 0.6 0.8 10
500
1000
1500
2000
Ellipse-90
0
Square-450
Circle
Square-00
Ellipse-00
Figure 11: Shape/fitting factor, , plotted as a function of porosity for different obstacle
shapes and orientations on square configurations, o45 . The straight dashed line
shows the value of 60 as in the original CK factor.
3.4. Corrections to the limit theories
Being unable to explain the variation of permeability with tortuosity and a constant shape
factor, now we attempt to optimize/correct the limit theories by Gebart [11] and
Drummond [9], see Table 1, in order to propose an analytical relation for the permeability
that is valid for all porosities and for square and hexagonal arrays of cylinders.
3.4.1 Square configuration
Assuming one particle at the center, pressure boundary at the left and right and periodic
top and bottom, we correct the original Gebart relation from Table 1, s
GK , by a linear
correction term 2
2
1
1
s s
G G
c
K Kg
, with g2=0.336. In contrast to s
GK , which
asymptotically approaches the limit case, c , but for 0.6 deviates already by
27
about 10%, the correction, KG2, has a maximum discrepancy in the range 0.85c of
less than 10%, and for 0.7c of less than 2%, see the squares in Fig. 12.
Since the Drummond relation from Table 1, s
DK , is valid at high porosities with
maximum discrepancy at 0.7 1 of less than 10% and for 0.8 1 of less than 2%,
we propose the following merged globally valid (for all porosities) function
2 2 ,s s s s
G D GK K K K m with 1 tanh /
,2
h tm
0.75, 0.037h t ,
that is valid for the whole range of porosity with deviations of less than 2% and it
includes the analytical relations for the limit cases.
0.2 0.4 0.6 0.8 10.6
0.7
0.8
0.9
1
1.1
FE
M / fit
KG
s [11]
Ks
G2
KD
s [9]
Tanh
Figure12: Relative error between FEM results and proposed corrections for square
configuration with the critical porosity 1 / 4c .
3.4.2 Staggered hexagonal configuration 060 :
In this situation, the correction to the Drummond relation from Table 1 is
2 1 21h h
D DK d K d , 2 ,h h h h
G D GK K K K m with d1=0.942, d2=0.153,
0.55, 0.037h t , leads to a corrected permeability for all porosities, with a
maximum error of less than 2%, see Fig. 13.
28
0.2 0.4 0.6 0.8 10.6
0.7
0.8
0.9
1
1.1
FE
M / fit
KG
h [11]
KD
h [9]
KD2
h
Tanh
Figure 13: Relative error between FEM results and proposed fits for hexagonal
configuration with the critical porosity 1 / 2 3c .
4. Summary and Conclusions
The permeability of porous structures is an important property that characterizes the
transport properties of porous media; however, its determination is challenging due to its
complex dependence on the microstructure of the media. Using an appropriate
representative volume element, transverse flow in aligned, periodic fibrous porous media
has been investigated based on high resolution (fine grid) FE simulations. This is
complementary to the recent study by Van der Hoef et. al. [30] who obtained the
drag/permeability relation for random arrays of monodisperse and bidisperse spheres. In
all of our simulations, the total pressure drop has been chosen small, such that we are
always in Darcy’s regime (creeping flow). In particular, the effects of different
parameters including fiber (particle) shape, aspect ratio, orientation, and staggered unit
29
cell angle on the normalized permeability are measured and discussed in detail for the full
range of porosities. The conclusions are:
The present results for the permeability are validated by comparing with available
theoretical and numerical data for square and hexagonal arrays over a wide range
of porosities. Especially in the limits of high and low porosity, agreement with
previous theoretical results is established.
By increasing the staggered unit cell angle, (where 60 degrees corresponds to
the hexagonal array), from the blocked configuration with minimal angle min, the
normalized permeability increases until it reaches its local maximum at 035 .
Then it decreases a bit (almost plateau) until it reaches its local minimum at 055 . From there it increases again until a maximum porosity is reached at
max. The best-fit (3rd
order) polynomials at different porosities are presented as
reference for later use.
By increasing the orientation angle of the ellipses (here the longer axis of ellipses
was used to define the orientation relative to the flow direction), the permeability
decreases and shows its anisotropy. The permeabilities at the extreme cases, i.e.,
eigenvalues at 00 and 90
0, are used to predict the permeability for arbitrary
orientation angles, see Eq. (9).
By increasing the aspect ratio of horizontal ellipses, the permeability increases
and approaches the permeability of slab flow, i.e., 2 /12 0.0833sK h [m
2] at
high porosities.
Using the hydraulic diameter concept the permeability can be expressed in the
general form of the (Carman Kozeny) CK equation. Our numerical results show
that the CK factor not only depends on the porosity but also on the pore structure,
namely particle shape, orientation and staggered angle .
The numerical results show that immobile circles and ellipses have the lowest and
highest permeability, respectively. The relevance of this observation for flows of
gas/fluid-solid with moving non-spherical particles is an open question.
Since analytical forms with the power as a free fit-parameter are neither consistent with
the highest porosity asymptote, nor with the lowest porosity limit case, those fits are only
an attempt to describe the intermediate regime of practical importance with a closed
functional form. In order to improve the analytical relation for the permeability, to be
applied, e.g., for DEM-FEM coupling, we proposed a merged function that includes both
30
limit cases of low and high porosity and is smooth in between with maximal deviation
from our numerical results of less than 2%.
Future work will investigate the creep and inertial flow regime through periodic and
disordered arrays, the relation between microstructure and (macro) permeability, and the
effect of the size of the system, especially for random/disordered structures. Since already
the packing generation algorithm affects the permeability in random arrays of parallel
cylinders, different procedures have to be compared and evaluated with respect to the
microstructure. These results can then be utilized for validation of advanced, more
coarse-grained models for particle-fluid interaction and their coupling-terms between the
discrete element method (DEM) for the particles and the FEM or CFD solver for the
fluid, in a multi-scale coarse grained approach.
Acknowledgements:
The authors thank N. P. Kruyt, A. Thornton and T. Weinhart for helpful discussion and
acknowledge the financial support of STW through the STW-MuST program, project
number 10120.
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